2026/05/25 by Zheyan Wan, Juven Wang, Shing-Tung Yau
#hep-th #cond-mat.str-el #hep-ph #math-ph #math.MP
Family puzzle asks why the Standard Model (SM) features exactly 3 families of quarks and leptons. Motivated by topological constraints, we study 4d fermionic anomalies with discrete Zn symmetry, classified by the 5d spin bordism group. We show that only the group-cohomology subclass H5(Zn,U(1))≅ Zn can be canceled by an anomalous Zn-symmetric 4d Zn-gauge topological quantum field theory (TQFT), while beyond-group-cohomology AZnp1 involving the Pontryagin class p1 cannot (except n=2,3). More generally, we prove that any cocycle αd∈Hd(Zn,U(1)) in odd spacetime dimension d≥3 is trivialised by the symmetry extension 1→ Zn→ Zn2→ Zn→ 1, and we construct the corresponding symmetric anomalous boundary TQFT. For d=5 and n=3, this yields a Spin× Z3-symmetric 4d Z3-gauge TQFT that cancels the mixed discrete (\bf B+L)-gauge-gravitational anomaly of the SM in the absence of 3 "sterile" right-handed neutrinos νR. We analyze a generalized SM with Nc colors and Nf families and argue that missing Nf copies of the νR can be naturally replaced by a 4d anomalous Spin×Z2FZ_2 Nf,\bf B + L symmetric ZN-gauge TQFT under the anomaly cancellation, via a ZN symmetry extension construction 1→ ZN→ Spin× ZNNf→ Spin×Z2FZ2Nf→1 of anomalous topological order. For minimal nonzero (N,Nf), the allowed minimal extensions are N=1,3,4,12, depending on divisibility of Nf by 2 and 3. Combining Witten anomaly and other constraints, we prove that N=Nc=Nf=3, with 3 families and 3 colors, is the unique minimal solution to match with the color-center baryon-to-quark symmetry extension 1→ ZNc→ Spin×Z2FZ_2NcNf,\bf Q+Nc\bf L→ Spin×Z2FZ_2 Nf,\bf B+LF→1. We also prove that AZ3p1=0\mod3 for the mod 3 cohomology class.